I dig as little as possible but I do have to on occasion, to remove the horrendous roots of Broussonetia papyrifera,the paper mulberry.
The problem with this tree, introduced in France in1750, is its tendency to propagate its horizontal roots at great distances from the tree itself
In fact there isn't a paper mulberry in the garden .There is one in next door's and several in the rough track leading to the garden but their roots are all invasive .Trees, even felled continue, to send out roots .
There is little made of this problem on official pages relating to this tree but a wealth of complaints on blogs and forums .One person claims (and I am willing to believe him/her) that the roots can tunnel under a road and show up on the other side .
If it weren't for extirpating as many of these roots as possible I would leave far more ground undisturbed .As it is I use a broad fork (grelinette) to aerate the ground and as soon as I encounter a maze of their orangey coloured roots commence to dig them up with a fork